Solving Ratio Problems with Marbles: A Practical Example

When working with problems involving ratios, it's important to break down the information provided and apply it systematically. This article will walk you through a detailed solution to a problem involving marbles: blue, green, and red. The steps will help you understand how to manipulate ratios and fractions, and how to apply them to find the desired ratio.

Problem Statement

Consider a bag containing blue, green, and red marbles. The ratio of the number of blue marbles to the number of green marbles is 2:5. Half of the total number of marbles is red. We need to find the ratio of the number of blue marbles to the number of red marbles.

Solution

Step 1: Define Variables

Let's define the variables as follows:

(n_b) number of blue marbles

(n_g) number of green marbles

(n_r) number of red marbles

Step 2: Apply Given Ratios and Information

From the problem, we know:

The ratio of blue marbles to green marbles is 2:5. This can be expressed as: (frac{n_b}{n_g} frac{2}{5}) This means we can express (n_b) in terms of (n_g): (n_b frac{2}{5}n_g)

We also know that half of the total number of marbles is red. Therefore:

(frac{1}{2}(n_b n_g) n_r) Multiplying both sides by 2 gives: (n_b n_g 2n_r) Substituting (n_b frac{2}{5}n_g) into the equation: (frac{2}{5}n_g n_g 2n_r) Rearranging this, we get: (frac{7}{5}n_g 2n_r) Solving for (n_r): (n_r frac{7}{10}n_g)

Step 3: Express Everything in Terms of a Common Variable

Now we can express everything in terms of (n_g):

(n_b frac{2}{5}n_g) (n_g n_g) (n_r frac{7}{10}n_g frac{7}{5} times frac{1}{2}n_g frac{7}{10}n_g)

Step 4: Find the Ratio of Blue Marbles to Red Marbles

The ratio of the number of blue marbles to the number of red marbles is:

(frac{n_b}{n_r} frac{frac{2}{5}n_g}{frac{7}{10}n_g} frac{2/5}{7/10} frac{2}{5} times frac{10}{7} frac{2 times 10}{5 times 7} frac{20}{35} frac{2}{7})

Conclusion

Therefore, the ratio of the number of blue marbles to the number of red marbles is:

(boxed{frac{2}{7}})

Problems Similar to This in a Nutshell:

Let the number of blue marbles be 2x. And that of green marbles be 5x. Now, the sum of numbers of blue and green marbles will be the half of the total number of marbles in the bag, which is 7x. It is also the number of red marbles in the bag. So, the ratio of blue marbles to red marbles is 2x / 7x 2/7. Hence, the ratio is 2:7. Let the number of blue, green, and red marbles be x, y, and z, respectively. Given ( frac{x}{y} frac{2}{5} ) and ( x y z ). Solving these equations, we get ( frac{x}{z} frac{2}{7} ). So the bag has three types of marbles: blue, green, and red. The ratio of blue to green is 2:5, and since half the total marbles are red, the ratio of blue to red is 2:7.

Conclusion

Understanding and solving problems involving ratios in the context of marbles can be very practical and engaging. By breaking down the problem and systematically applying the given information, we can easily find the desired ratio. This method can be applied to various real-world scenarios involving ratios and proportions, making it a valuable skill to master.